Lesson 46 Chapter 9: Angles and Shapes Quadrilaterals A quadrilateral is any four-sided shape. Any quadrilateral can be split up into two triangles by drawing in a diagonal, like this: The sum of the four angles in any quadrilateral is 360 o i.e. w + x + y + z = 360 o You can find this by just splitting the quadrilateral into two triangles. i.e. ABC = 180 o and ADC = 180 o A So the sum of the two triangles is equal to 180 o + 180 o = 360 o x w z y D which is equal to the sum of the angles of a quadrilateral since these two triangles form a quadrilateral. Types of Quadrilaterals B C There are special types of quadrilateral : the rectangle the rhombus the square (those are all parallelograms), and there is also: the trapezoid the kite If it isn't one of those it is an irregular quadrilateral. 9 Angles & Shapes Page 1
Here are the details: The Rectangle means "right angle" and show equal sides A rectangle is a four-sided shape where every angle is a right angle (90 ). Also opposite sides are parallel and of equal length. The Rhombus A rhombus is a four-sided shape where all sides have equal length. Also opposite sides are parallel and opposite angles are equal. Another interesting thing is that the diagonals (dashed lines in second figure) of a rhombus bisect each other at right angles. The Square means "right angle" show equal sides A square has equal sides and every angle is a right angle (90 ) Also opposite sides are parallel. A square also fits the definition of a rectangle (all angles are 90 ), and a rhombus (all sides are equal length). 9 Angles & Shapes Page 2
The Parallelogram Opposite sides are parallel and equal in length, and opposite angles are equal (angles "a" are the same, and angles "b" are the same) NOTE: Squares, Rectangles and Rhombuses are all Parallelograms! Example: if a parallelogram has all sides equal and angles "a" and "b" are right angles, then it is also a square. The Trapezoid (or Trapezium) Trapezoid Regular Trapezoid A trapezoid (UK: trapezium) has one pair of opposite sides parallel. It is called a regular trapezoid if the sides that aren't parallel are equal in length and both angles coming from a parallel side are equal, as shown. A trapezoid is not a parallelogram because only one pair of sides is parallel. The Kite Hey, it looks like a kite. It has two pairs of sides. Each pair is made up of adjacent sides that are equal in length. The angles are equal where the pairs meet. Diagonals (dashed lines) meet at a right angle, and one of the diagonal bisects (cuts equally in half) the other.... and that's it for the special quadrilaterals; if it doesn't fit one of those it is an Irregular Quadrilateral: 9 Angles & Shapes Page 3
Irregular Quadrilaterals Polygon Any quadrilateral that doesn't match one of the previous types. A quadrilateral is a polygon. In fact it is a 4-sided polygon, just like a triangle is a 3-sided polygon, a pentagon is a 5-sided polygon, and so on. Example 1 Work out the value of x. Answer Rule: The sum of interior angles of a quadrilateral is equal to 360 o. o 136 45 o x o So so x + 136 o + 45 o + 100 o = 360 o x + 281 o = 360 o x = 360 o 281 o = 79 o 100 o 9 Angles & Shapes Page 4
Exercise 1 Find the angles of each marked angle and give an appropriate reason. 1) 2) 112 o 124 o t 80 o y s 70 o Ans. y = Reason: Ans. t = s = Reason: 3) 82 o 135 o Reason: Ans. a = Reason: Ans. b = Reason: 124 o a b c Ans. c = Reason: 4) 133 o x Ans. x = Reason: 100 o y Ans. y = Reason: 5) g 120 o Ans. g = Reason: 70 o h j i Ans. h = Reason: Ans. i = Reason: Ans. j = Reason: 9 Angles & Shapes Page 5
6) f c Ans. c = Reason: d e 105 o Ans. d = Reason: Ans. e = Reason: Ans. f = Reason: 7) 116 o Ans. c = Reason: e d c 55 o Ans. d = Reason: Ans. e = Reason: 8) 87 o k l Ans. j = Reason: Ans. k = Reason: 58 o 73 o j Ans. l = Reason: 9 Angles & Shapes Page 6
Exercise 2 1) 1 2 3 4 5 6 a) Write down the special name for each of these four-sided shapes. b) For each shape say which statements in the list below are true. i. The sides are equal ii. Both pairs of opposite sides are parallel iii. Only opposite angles are equal iv. There is just one pair of parallel sides v. The four sides are made up of just two distinct lengths. vi. The diagonals make an angle of 90 o where they meet. 2. In the following questions, some of the diagrams contain more than one quadrilateral. Name each quadrilateral with the appropriate letters and find the size of each marked angle. a) b) s o p o 35 o x o 86 o z o r o t o q o 57 o y o 9 Angles & Shapes Page 7
c) d) z o 66 o q o p o 25 o y o x o r o 50 o 3. A 3x o B 4x o D 2x o x o C a) Find the value of x. b) Prove that ABCD is a trapezium. 4. ABCD is a trapezium with AB parallel to DC. A = 127 o and C =76 o. Work out B and D. (Hint: draw a diagram first.) 5. JKLM is a rectangle. JK is the longer side. N is a point on JK such that KN = KL. Given MNL = 79 o work out JMN. 6. PQRS is a rhombus with the diagonal QS equal to PQ. Work out P. 7. ABCD is a rectangle. E is a point on AB. a) What type of quadrilateral is BCDE? b) Given ADE = 50 o work out DEB. c) If DE = EB work out EBD. 9 Angles & Shapes Page 8